Solve 3-4/3+sqrt{5}

$$3 - \frac { 4 } { 3 + \sqrt { 5 } }$$

Evaluate

$$\sqrt{5}\approx 2.236067977$$

Solution Steps

Rationalize the denominator of \frac{4}{3+\sqrt{5}} by multiplying numerator and denominator by 3-\sqrt{5}.

$$3-\frac{4\left(3-\sqrt{5}\right)}{\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)}$$

Consider \left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$3-\frac{4\left(3-\sqrt{5}\right)}{3^{2}-\left(\sqrt{5}\right)^{2}}$$

Square 3. Square \sqrt{5}.

$$3-\frac{4\left(3-\sqrt{5}\right)}{9-5}$$

Subtract 5 from 9 to get 4.

$$3-\frac{4\left(3-\sqrt{5}\right)}{4}$$

Cancel out 4 and 4.

$$3-\left(3-\sqrt{5}\right)$$

To find the opposite of 3-\sqrt{5}, find the opposite of each term.

$$3-3-\left(-\sqrt{5}\right)$$

The opposite of -\sqrt{5} is \sqrt{5}.

$$3-3+\sqrt{5}$$

Subtract 3 from 3 to get 0.

$$\sqrt{5}$$